sciencefandomcom_el-20200214-history
Διανυσματική Ανάλυση
Διανυσματική Ανάλυσις Vector Analysis, Διανυσματικός Λογισμός - Επιστημονικός Κλάδος των Μαθηματικών Ετυμολογία Η ονομασία "Διανυσματική" σχετίζεται ετυμολογικά με την λέξη "διάνυσμα". Εισαγωγή Basic objects The basic objects in vector calculus are scalar fields (scalar-valued functions) and vector fields (vector-valued functions). These are then combined or transformed under various operations, and integrated. In more advanced treatments, one further distinguishes pseudovector fields and pseudoscalar fields, which are identical to vector fields and scalar fields except that they change sign under an orientation-reversing map: for example, the curl of a vector field is a pseudovector field, and if one reflects a vector field, the curl points in the opposite direction. This distinction is clarified and elaborated in geometric algebra, as described below. Vector operations Algebraic operations The basic algebraic (non-differential) operations in vector calculus are referred to as vector algebra, being defined for a vector space and then globally applied to a vector field, and consist of: ;scalar multiplication: multiplication of a scalar field and a vector field, yielding a vector field: a \bold{v} ; ;vector addition: addition of two vector fields, yielding a vector field: \bold{v}_1 + \bold{v}_2 ; ;dot product: multiplication of two vector fields, yielding a scalar field: \bold{v}_1 \cdot \bold{v}_2 ; ;cross product: multiplication of two vector fields, yielding a vector field: \bold{v}_1 \times \bold{v}_2 ; There are also two triple products: ;scalar triple product: the dot product of a vector and a cross product of two vectors: \bold{v}_1\cdot\left( \bold{v}_2\times\bold{v}_3 \right) ; ;vector triple product: the cross product of a vector and a cross product of two vectors: \bold{v}_1\times\left( \bold{v}_2\times\bold{v}_3 \right) or \left( \bold{v}_3\times\bold{v}_2\right)\times\bold{v}_1 ; although these are less often used as basic operations, as they can be expressed in terms of the dot and cross products. Differential operations Vector calculus studies various differential operators defined on scalar or vector fields, which are typically expressed in terms of the del operator ( \nabla ), also known as "nabla". The four most important differential operations in vector calculus are: where the curl and divergence differ because the former uses a cross product and the latter a dot product, and f'' denotes a scalar field and '''F' denotes a vector field. A quantity called the Jacobian is useful for studying functions when both the domain and range of the function are multivariable, such as a change of variables during integration. Theorems Likewise, there are several important theorems related to these operators which generalize the fundamental theorem of calculus to higher dimensions: Θέματα - Τομείς *Διανυσματικό Πεδίο. *Κλίση, * Απόκλιση, *Περιστροφή. *Στροβιλισμός *Ανάδελτα. *Λαπλασιανή *Συντηρητικό Πεδίο και ανεξαρτησία από τον δρόμο ολοκλήρωσης. *Προσδιορισμός της δυναμικής συνάρτησης. *Θεωρήματα του Green, Gauss και Stokes και οι εφαρμογές τους. *Αναλλοίωτη μορφή του ανάδελτα. *Σωληνοειδές Πεδίο. *Προσδιορισμός της διανυσματικής συνάρτησης. *Ολοκληρωτικοί τύποι, του Green, κατά grad, κατά rot. *Φυσική ερμηνεία της απόκλισης και της περιστροφής. *Φυσικές εφαρμογές. Υποσημειώσεις Εσωτερική Αρθρογραφία *Ηλεκτρικό Πεδίο *Ηλεκτρικό Φορτίο *Δονούμενο Ηλεκτρικό Κύκλωμα Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] Category: Απειροστική Ανάλυση Category: Μαθηματικά Category: Επιστήμες